6-8 duoprism

In geometry of 4 dimensions, a 6-8 duoprism, a duoprism and 4-polytope resulting from the Cartesian product of a hexagon and an octagon.

6-8 duoprism
TypePrismatic uniform polychoron
Schläfli symbol{6}×{8}
{6}×t{4}
t{3}×{8}
t{3}×t{4}
Coxeter diagram


Cells6 octagonal prisms,
8 hexagonal prisms
Faces48 squares
6 octagons
8 hexagons
Edges96
Vertices48
Vertex figureDigonal disphenoid
Symmetry[6,2,8], order 192
Dual6-8 duopyramid
Propertiesconvex, vertex-uniform

Images
Schlegel diagrams
Image
Center Octagonal prism Hexagonal prism

6-8 duopyramid
6-8 duopyramid
Typeduopyramid
Schläfli symbol{6}+{8}
{6}+t{4}
t{3}+{8}
t{3}+t{4}
Coxeter-Dynkin diagram


Cells48 digonal disphenoids
Faces96 isosceles triangles
Edges62 (48+6+8)
Vertices14 (6+8)
Symmetry[6,2,8], order 192
Dual6-8 duoprism
Propertiesconvex, facet-transitive

The dual of a 6-8 duoprism is called a 6-8 duopyramid. It has 48 digonal disphenoid cells, 96 isosceles triangular faces, 62 edges, and 14 vertices.


The 3-4 duoantiprism is an alternation of the 6-8 duoprism, but is not uniform. It has a highest symmetry construction of order 96, with 40 cells composed of 6 square antiprisms, 8 octahedra (as triangular antiprisms), and 24 tetrahedra (as digonal disphenoids). There exists a construction with uniform square antiprisms with an edge length ratio of 1 : 1.456, and also with regular octahedra with an edge length ratio of 0.663 : 1.


Vertex figure for the 3-4 duoantiprism

Also related is the bialternatosnub 3-4 duoprism, constructed by removing alternating long rectangles from the octagons, but is also not uniform. It has a highest symmetry construction of order 48, with 6 rectangular trapezoprisms (topologically equivalent to a cube but with D2d symmetry), 4 octahedra (as triangular antiprisms), 4 triangular prisms (both from the hexagonal prisms), with 24 triangular prisms (as C2v-symmetry wedges) filling the gaps. Its vertex figure is a Cs-symmetric polyhedron formed by augmenting a tetrahedron on one of the square pyramid's triangular faces.


Vertex figure for the bialternatosnub 3-4 duoprism

See also

Notes

    References
    • Regular Polytopes, H. S. M. Coxeter, Dover Publications, Inc., 1973, New York, p. 124.
    • John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 26)
    • Norman Johnson Uniform Polytopes, Manuscript (1991)
      • N. W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
    • Catalogue of Convex Polychora, section 6, George Olshevsky.
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